Does average mean something like "as expected"?

[Agent Smith]

TL;DR Summary

Does average mean something like "as expected"?

A conducts an experiment. I don't know how to describe this experiment, but let's say he measures the masses of $3$ things and computes their sum. He finds that the mean mass equals $2$ units of mass. He considers this to be special in some sort of way, something that demands an explanation.

B finds out that the the masses of these $3$ things can take on the values $1, 2, 3$. He reasons statistically that $\mu_{mass} = \frac{1 + 2 + 3}{3} = 2$ and so What A found is something one would describe as "it is as expected".

How would you respond to this?

 

[FactChecker]

IMO, the term "as expected" does not have a formal mathematical meaning. Suppose that you knew a continuous random variable has a mean of 2 and a standard deviation of 0.5. Now suppose that a sample of 3 had an average of 2.0000000000000. I would not call that "as expected". I would start looking for some trick.

 

[phinds]

How would you respond to this?

"As expected" is, as @FactChecker pointed out, not a scientific term and more importantly, something is "as expected" (English language usage) only if one has some sort of baseline to create an expectation. With no further information other than what is given, "as expected" is meaningless.

I mean, suppose the actual normal average values for the three things being measured are 3, 8, and 10. In that case you could hardly call 2 "as expected".

 

[anuttarasammyak]

I read OP that he/she has some question about expected value, ref. https://en.wikipedia.org/wiki/Expected_value , and its relation with "average". I myself have no idea how to distinguish them clearly.

 

[FactChecker]

The mathematical meaning of "expected value" is formally defined and it is very different from the English use of "expected" in "as expected". Suppose there is a random variable based on coin tosses which gives values of -100 and 100 with equal probability of 1/2. The expected value is 0. But if that random variable ever gave an actual value of 0, that would not be "as expected". In fact, it would be impossible.

 

[gleem]

Does average mean something like "as expected"?

The expected value of a measurement is based on a hypothesis. The expected value is believed to be in the interval defined by the average and the uncertainly (standard deviation) of the measurements. We might say that the average "agrees" with the expected value.

Your example here is not a realistic scenario for a discussion without an estimate of the uncertainties. What was A's standard deviation? B's statistical would be 0.7 if one thinks his data makes sense.

 

[Agent Smith]

IMO, the term "as expected" does not have a formal mathematical meaning. Suppose that you knew a continuous random variable has a mean of 2 and a standard deviation of 0.5. Now suppose that a sample of 3 had an average of 2.0000000000000. I would not call that "as expected". I would start looking for some trick.

We do use the word "expected" in math, most relevantly the mean is aka expected value.

I read OP that he/she has some question about expected value, ref. https://en.wikipedia.org/wiki/Expected_value , and its relation with "average". I myself have no idea how to distinguish them clearly.

Gracias for the link. As far as I can tell, the arithmetic mean is expected value.

"As expected" is, as @FactChecker pointed out, not a scientific term and more importantly, something is "as expected" (English language usage) only if one has some sort of baseline to create an expectation. With no further information other than what is given, "as expected" is meaningless.

I mean, suppose the actual normal average values for the three things being measured are 3, 8, and 10. In that case you could hardly call 2 "as expected".

I don't know if "baseline" is the correct word, but I think the fact that $\frac{1 + 2 + 3}{3} = 2$ gives us an idea of what we should expect from A's experiment, no?

 

[phinds]

I don't know if "baseline" is the correct word, but I think the fact <eqn didn't xfer> that gives us an idea of what we should expect from A's experiment, no?

No. Read post #3 again.

 

[FactChecker]

We do use the word "expected" in math, most relevantly the mean is aka expected value.

"expected value" has a very precise mathematical meaning and has to be used carefully. "expected" does not have the same meaning without "value".

Gracias for the link. As far as I can tell, the arithmetic mean is expected value.

The sample mean is not the same as the "expected value" of a random distribution. The sample mean is an estimate of the expected value of a random variable. But the sample mean (especially for small samples) may be very different from the expected value. In fact, the expected value may be a number (or even infinity) that a finite sample mean can never have.

 

[Agent Smith]

Thank you all.